Abstract
An intriguing phenomenon in many instances of compressed sensing is that the reconstruction quality is governed not just by the overall sparsity of the object to recover, but also on its structure. This chapter is about understanding this phenomenon, and demonstrating how it can be fruitfully exploited by the design of suitable sampling strategies in order to outperform more standard compressed sensing techniques based on random matrices.
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Notes
- 1.
In actual fact, the sensing device takes measurements of the continuous Fourier transform of a function \(x \in L^{2}(\mathbb{R}^{d})\). As discussed in [1, 4], modelling continuous Fourier measurements as discrete Fourier measurements can lead to inferior reconstructions, as well as inverse crimes. To avoid this, one must consider an infinite-dimensional compressed sensing approach, as in (5.2). See [2, 4] for details, as well as [29] for implementation in MRI. For simplicity, we shall continue to work with the finite-dimensional model in the remainder of this chapter.
- 2.
- 3.
For a proof, we refer to [3].
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Acknowledgements
The authors thank Andy Ellison from Boston University Medical School for kindly providing the MRI fruit image, and General Electric Healthcare for kindly providing the brain MRI image. BA acknowledges support from the NSF DMS grant 1318894. ACH acknowledges support from a Royal Society University Research Fellowship. ACH and BR acknowledge the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/L003457/1.
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Adcock, B., Hansen, A.C., Roman, B. (2015). The Quest for Optimal Sampling: Computationally Efficient, Structure-Exploiting Measurements for Compressed Sensing. In: Boche, H., Calderbank, R., Kutyniok, G., VybĂral, J. (eds) Compressed Sensing and its Applications. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-16042-9_5
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